Insights into the Zonal-Mean Response of the Hydrologic Cycle to GlobalWarming from a Diffusive Energy Balance Model

NICHOLAS SILER

College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

GERARD H. ROE

Department of Earth and Space Sciences, University of Washington, Seattle, Washington

KYLE C. ARMOUR

School of Oceanography and Department of Atmospheric Sciences, University of Washington, Seattle, Washington

(Manuscript received 13 February 2018, in final form 25 May 2018)

ABSTRACT

Recent studies have shown that the change in poleward energy transport under global warming is well

approximated by downgradient transport of near-surface moist static energy (MSE) modulated by the spatial

pattern of radiative forcing, feedbacks, and ocean heat uptake. Here we explore the implications of down-

gradientMSE transport for changes in the vertically integrated moisture flux and thus the zonal-mean pattern

of evaporation minus precipitation (E 2 P). Using a conventional energy balance model that we have

modified to represent theHadley cell, we find that downgradientMSE transport implies changes inE2P that

mirror those simulated by comprehensive global climate models (GCMs), including a poleward expansion of

the subtropical belt whereE. P, and a poleward shift in the extratropical minimum ofE2 P associated with

the storm tracks. The surface energy budget imposes further constraints on E and P independently: E in-

creases almost everywhere, with relatively little spatial variability, while P must increase in the deep tropics,

decrease in the subtropics, and increase in middle and high latitudes. Variations in the spatial pattern of

radiative forcing, feedbacks, and ocean heat uptake across GCMs modulate these basic features, accounting

for much of the model spread in the zonal-mean response ofE and P to climate change. Thus, the principle of

downgradient energy transport appears to provide a simple explanation for the basic structure of hydrologic

cycle changes in GCM simulations of global warming.

1. Introduction

In the zonal and annual mean, Earth’s atmosphere

transports energy down the meridional insolation gra-

dient, from the tropics to higher latitudes. Consequently,

the climate is much more temperate than it would be if

radiative equilibrium were imposed at each latitude

(e.g., Hartmann 1994; Pierrehumbert 2010).

The dominant mechanisms of atmospheric energy

transport vary by region. Outside the tropics, most

poleward energy transport is accomplished by mid-

latitude eddies (both stationary and transient), which

transport moist, warm, tropical air poleward and cool,

dry, polar air equatorward. In the tropics, by contrast,

poleward energy transport is primarily accomplished by

the Hadley cell, whose total energy transport is a small

residual of offsetting contributions from its lower

(equatorward) and upper (poleward) branches (e.g.,

Hartmann 1994). Because the tropical atmosphere is

weakly stable, moist static energy increases with height,

and thus energy transport is slightly larger within the

upper branch of the Hadley cell, resulting in poleward

transport overall. This constraint has proven funda-

mental to understanding important aspects of the Had-

ley cell’s behavior, including its seasonality (e.g.,

Donohoe et al. 2013) and its hemispheric asymmetry as a

consequence of cross-equatorial ocean heat transport

(Frierson et al. 2013; Marshall et al. 2014).

Atmospheric energy transport is also closely tied to

the hydrologic cycle, as shown in Fig. 1. One form of

atmospheric energy transport is the latent heat ofCorresponding author: Nicholas Siler, nick.siler@oregonstate.

edu

15 SEPTEMBER 2018 S I L ER ET AL . 7481

DOI: 10.1175/JCLI-D-18-0081.1

� 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS CopyrightPolicy (www.ametsoc.org/PUBSReuseLicenses).

moisture, the divergence of which is proportional to

evaporation minus precipitation at the surface (E 2 P;

black line in Fig. 1). Outside the tropics, latent heat is

transported downgradient by eddies, resulting in net

divergence (and thus E 2 P . 0) in the subtropics and

net convergence (and thus E 2 P , 0) at higher lati-

tudes. In contrast, the Hadley cell transports latent heat

upgradient as a result of equatorward flow at low levels,

where water vapor is concentrated. Such upgradient

transport leads to strong convergence of latent heat near

the equator [known as the intertropical convergence

zone (ITCZ)], and adds to the latent heat divergence in

the subtropics. Because E is constrained by net surface

radiation, which has relatively little spatial structure, it

varies weakly with latitude (Fig. 1, red line). The un-

dulations in E 2 P are thus mirrored in P (Fig. 1, blue

line), explaining why deserts are largely confined to the

subtropics, while the deep tropics are home to many of

the wettest places on Earth.

The principle of downgradient energy transport by the

atmosphere has given rise to the widespread use of en-

ergy balance models (EBMs) to study the zonal-mean

climate. The theoretical basis for EBMs comes from the

fact that, on long time scales, the zonal-mean net heating

of the atmosphere Qnet must be balanced by the di-

vergence of northward column-integrated atmospheric

energy transport F:

Qnet(x)5

1

2pa2dF

dx, (1)

where a is the radius of Earth, x is the sine of latitude,

and Qnet is the difference between the net downward

energy flux at the top of the atmosphere (TOA) and the

surface (e.g., Pierrehumbert 2010). In most early EBMs

(Budyko 1969; North 1975), F was assumed to be pro-

portional to the meridional gradient in near-surface

temperature T. However, recognizing that T represents

only the sensible component of atmospheric heat con-

tent, more recent EBM formulations have replaced T

with near-surface moist static energy h 5 cpT 1 Lq,

where cp is the specific heat of air, L is the latent heat of

vaporization, and q is the near-surface specific humidity1

(e.g., Flannery 1984; Frierson et al. 2007; Hwang and

Frierson 2010; Rose et al. 2014; Roe et al. 2015). In the

continuous limit, energy transport in the moist static

EBM is governed by Fickian diffusion

F(x)522pp

s

gD(12 x2)

dh

dx, (2)

where ps is surface air pressure (1000 hPa), g is the ac-

celeration due to gravity, D is a (constant) diffusion

coefficient (with units of m2 s21), and (1 2 x2) accounts

for the spherical geometry. Typically, relative humidity

is assumed to be fixed at 80%, making h a single-valued

function of T (Hwang and Frierson 2010; Rose et al.

2014; Roe et al. 2015). While unrealistic over land, this

value is close to the observed near-surface relative hu-

midity over the oceans, where most eddy heat transport

occurs (Peixoto and Oort 1996). Finally, combining Eqs.

(1) and (2) yields a single expression for the moist static

energy balance model (MEBM) that relates Qnet di-

rectly to h (and thus T):

Qnet(x)52

ps

ga2D

d

dx

�(12 x2)

dh

dx

�. (3)

The MEBM defined in Eq. (3) has proven to be re-

markably successful at emulating the climate response

to anthropogenic forcing simulated by comprehensive

global climate models (GCMs). For example, Hwang

and Frierson (2010) showed that the MEBM success-

fully predicts the increase in midlatitude poleward heat

transport with global warming, and found that the in-

termodel differences in heat transport can be explained

by the MEBM as a result of differences in climate

feedbacks. More recent studies have also found that the

MEBM realistically emulates the surface temperature

response to different spatial patterns of ocean heat up-

take (Rose et al. 2014) and climate feedbacks (Roe et al.

2015) across a range of GCMs.

FIG. 1. TheE2P (black line) fromERA-Interim, averaged over

the 35-yr period from 1981 to 2010 (scaled by L to have units of

Wm22), also E (red line) and P (blue line). Red shading indicates

subtropical latitudes where E . P, implying a net divergence of

atmospheric latent heat transport. Blue shading indicates latitudes

where P. E, which includes the ITCZ. The profile of E2 P is the

result of poleward latent heat transport by midlatitude eddies

(indicated schematically in purple) and equatorward latent heat

transport within the lower branch of the HC (green).

1 Note that this definition of h neglects geopotential, which is

close to zero near the surface.

7482 JOURNAL OF CL IMATE VOLUME 31

Despite these successes, the MEBM has an important

limitation: it does not correctly simulate latent heat

transport, which is upgradient within the Hadley cell. To

address this limitation, here we incorporate a simple

Hadley cell parameterization into the MEBM, which

allows latent heat to travel upgradient even while total

energy transport is downgradient, thus permitting a

more realistic representation of the hydrologic cycle. In

section 2, we show that the modified MEBM is able to

reproduce the basic pattern of E 2 P in the climatology

of both observations and climate models. In section 3,

we use the modified MEBM to study the zonal-mean

response of E 2 P to global warming, giving special

attention to how the MEBM solution differs from the

well-known thermodynamic scaling argument of Held

and Soden (2006). We show that the MEBM captures

important aspects of the change in E 2 P in GCM

simulations, including the poleward expansion of the

subtropics and the poleward shift in the midlatitude

storm tracks. When driven by the spatial patterns of

forcing, feedbacks, and ocean heat uptake correspond-

ing to individual GCMs, the MEBM also captures much

of the intermodel variability in the response of E2 P to

climate change. Finally, using our recent formulation for

the dependence of evaporation on surface temperature

and ocean heat uptake (Siler et al. 2018), we disaggre-

gate the individual contributions of P and E, and show

that each reproduces many of the features seen in cli-

mate models.

2. The modified MEBM and its climatology

Figure 2a shows annual-meanQnet(x) calculated from

ERA-Interim over the period 1981–2010. From Eq. (1),

positive values of Qnet in the tropics indicate net di-

vergence of atmospheric heat transport, while negative

values at higher latitudes indicate net convergence. In-

serting this Qnet profile into Eq. (3), we then choose a

value of D so that T(x) resembles the observed zonal-

mean 2-m air temperature at sea level. The best agree-

ment is found by settingD5 1.163 106m2 s21 (Fig. 2b),

which is within 10%of the value used in previous studies

(Hwang and Frierson 2010; Rose et al. 2014; Roe et al.

2015). Like these studies, we assume that D is uniform

with latitude and held fixed at this value for all following

analyses.

To give the MEBM a realistic hydrologic cycle, we

must alter the partitioning between latent and dry static

heat transport within the tropics, where latent heat

transport is upgradient. This idea originated with Sellers

(1969), whose primitive 10-box EBM included the ad-

vection of latent and sensible heat by the mean meridi-

onal wind, which he fit to observations.We seek a less ad

hoc approach, using Eqs. (1)–(3) and a few additional

constraints to solve for the meridional circulation and its

associated latent heat transport.

Our first step is to use a weighting function w(x) to

partition the total heat transport into Hadley cell (HC)

and eddy components

FHC

(x)5w(x)F(x) and (4)

Feddy

(x)5 [12w(x)]F(x) . (5)

For w(x), we choose a Gaussian function

w(x)5 exp

�2x2

s2x

�, (6)

and specify a characteristic width of sx5 0.3, resulting in

the profile shown in Fig. 2c. With this choice, eddies

account for essentially all energy transport poleward of

458 latitude, while the Hadley cell accounts for most

energy transport inside of 158 latitude (and 100% at the

equator). Note that the original MEBM is recovered by

setting w(x) 5 0 (dashed red line in Fig. 2c).

There are two obvious limitations to Eq. (6). First,

because w(x) approaches zero outside the tropics, it

does not account for heat transport by the Ferrel and

polar cells, which comprise the extratropical compo-

nents of the mean meridional circulation. Second, by

choosing a fixed value for sx, we neglect possible

changes in the partitioning between eddy and Hadley

cell transport in response to global warming. In efforts to

address these limitations, we have experimented with

various other representations of w(x), but have gener-

ally found that any gains in realism come with increased

complexity, and do not change the essence of the climate

response. In our view, Eq. (6) seems to strike an ap-

propriate balance between simplicity and realism.

Having quantified the total energy transported by the

Hadley cell [Eq. (4)], its circulation can then be de-

termined following Held (2001). Let c(x) be the south-

ward mass transport in the lower branch of the cell,

which by mass conservation is equal to the northward

mass transport in the upper branch. Then

FHC

(x)5c(x)g(x) , (7)

where g(x) is the gross moist stability of the atmosphere,

which represents the flow-weighted difference in moist

static energy between the upper and lower branches at

each latitude (Neelin and Held 1987). In the tropical

upper troposphere, temperature gradients tend to be

small, and moist static energy is relatively uniform. As a

result, variations in g(x) are primarily caused by varia-

tions in h(x), implying that

15 SEPTEMBER 2018 S I L ER ET AL . 7483

g(x)’ hT2h(x) , (8)

where hT represents the (spatially uniform) moist static

energy in the tropical upper troposphere (Held 2001;

Merlis et al. 2013; Hill et al. 2015). We set hT 5 1.06 3h(0), or 6% above the near-surfacemoist static energy at

the equator, which results in the best agreement with

observed moisture transport, and is a reasonable ap-

proximation of the equatorial moist static energy profile

in reanalysis. With hT specified, we then solve for c(x)

from Eqs. (4) and (7) (Fig. 2d).

Since g(0). 0, the net transport of moist static energy

is downgradient throughout the Hadley cell. However,

because the upper branch of the cell is essentially dry,

latent heat transport Fq is confined to the lower branch,

implying that

FHC,q

(x)52c(x)Lq(x) . (9)

Combining the latent heat transport by the Hadley cell

with the downgradient eddy contribution in the extra-

tropics gives the total latent heat transport Fq(x)

(Fig. 2e). Finally, the divergence of Fq(x) gives E 2 P

(Fig. 2f).

Figure 3 shows the same MEBM solutions for T and

E 2 P as in Figs. 2b and 2f, but now based on the av-

erage Qnet profile among GCM simulations of the

preindustrial climate from the latest Coupled Model

FIG. 2. (a) TheQnet calculated fromERA-Interim over the period 1981–2010, which is equal to the net downward

radiative flux at the TOAminus ocean heat uptake. The x axis is area-weighted, and is therefore linear in x, the sine

of latitude. (b) Zonal-mean air temperature at sea level from ERA-Interim (blue) and the MEBM (red), withD51.16 3 106m2 s21. (c) A weighting function representing the fraction of total atmospheric energy transport in the

MEBM that is performed by the HC. The dashed red line at w 5 0 shows the weighting function implicit in the

original MEBM with no HC. (d) The southward mass transport within the lower branch of the mean meridional

circulation in ERA-Interim (blue line), the modified MEBM (solid red line), and the original MEBM (dashed red

line). In ERA-Interim, this value applies to the layer below 900 hPa, where most atmospheric water vapor is

concentrated, and includes contributions from the Ferrel and polar cells in addition to the HC. (e) As in (d), but for

the northward latent heat transport by the atmosphere. (f) As in (d), but forE2 P, which is equal to the divergence

of northward latent heat transport in (e).

7484 JOURNAL OF CL IMATE VOLUME 31

Intercomparison Project (CMIP5). The variables in

Fig. 3 were computed exactly as described above (i.e.,

with the same value of D and Hadley cell parameters),

so that the differences between Figs. 2 and 3 are entirely

due to differences in Qnet. In both Figs. 2 and 3, dashed

red lines show theMEBM solution without aHadley cell

[i.e., w(x) 5 0].

Figures 2 and 3 show that the Hadley cell version of

the MEBM is a significant improvement over the origi-

nal MEBM. Based on a simple diffusive representation

of downgradient energy transport, it successfully emu-

lates the basic structure of the zonal-mean hydrologic

cycle in both reanalysis and GCM simulations. In par-

ticular, the MEBM captures the minima in E 2 P as-

sociated with the ITCZ and midlatitude storm tracks,

and also provides a reasonable estimate of the latitudes

where E 2 P 5 0, which mark the boundaries of the

subtropics (Fig. 1). And despite some discrepancies in

E 2 P in the tropics, the MEBM correctly produces the

double ITCZ evident in CMIP5 models, and the single

ITCZ north of the equator in observations. This result

adds further support to the growing consensus that

hemispheric asymmetries in Qnet, and required cross-

equatorial heat transport, play a leading role in setting

the location of the ITCZ (Kang et al. 2008; Frierson and

Hwang 2012; Frierson et al. 2013; Hwang and Frierson

2013; Hwang et al. 2013; Donohoe et al. 2013, 2014;

Marshall et al. 2014).

A summary of energy transports in the modified

MEBM is given in Table 1, with columns representing

the eddy (Feddy) and Hadley cell (FHC) contributions

and rows representing the latent heat (Fq) and dry static

energy (FT) contributions. The sum of all four terms is

equal to the total energy transport, which is down-

gradient and set by Eq. (2) with constant diffusivity and

relative humidity. In the extratropics, where eddy trans-

port dominates (i.e.,w’ 0), Fq and FT are both diffusive.

In the tropics, however, Fq is upgradient within the

lower branch of the Hadley cell. We have shown that

this modification allows the MEBM to reproduce the

broad structure of the zonal-mean hydrologic cycle in

bothGCMs and reanalysis. In the next section, we use the

modified MEBM to explore the implications of down-

gradient energy transport for how the zonal-mean hy-

drologic cycle responds to global warming.

3. Global warming simulations

In section 5 of their seminal paper on the response of

the hydrologic cycle to global warming, Held and Soden

(2006, hereinafter HS06) consider the implications of

downgradient energy transport for the change in zonal-

mean E 2 P. Conceptualizing the transport response as

diffusive, they argue that, because of relatively weak

gradients in the spatial pattern of warming, the pattern

of E 2 P should amplify with surface warming at a rate

roughly equal to

E0 2P0

E2P’aT 0 , (10)

where primes indicate the difference between the

warmed and mean-state climates and

a5L

RyT2

(11)

is the Clausius–Clapeyron scaling factor, with Ry rep-

resenting the gas constant of water vapor.

FIG. 3. As in Figs. 2b,f, but for the ensemble mean of 20 CMIP5 simulations of the preindustrial climate (blue line).

The MEBM solution was computed exactly as for Fig. 2, but using the Qnet profile from the CMIP5 ensemble.

TABLE 1. (top) Northward latent and (bottom) dry static en-

ergy transport by the (left) eddies and (right) HC in the MEBM.

The sum of all four terms is equal to the total energy transport F(x)

[Eq. (2)].

Eddies Hadley cell

Fq(x) 22pps

gD(12 x2)L

dq

dx[12w(x)] 2c(x)Lq(x)

FT(x) 22pps

gD(12 x2)cp

dT

dx[12w(x)] c(x)[g(x)1Lq(x)]

15 SEPTEMBER 2018 S I L ER ET AL . 7485

HS06 found that Eq. (10) broadly captured the mag-

nitude and spatial pattern of E0 2 P0 predicted in GCM

simulations of global warming. However, they also

identified robust aspects of E0 2 P0 in GCMs that Eq.

(10) did not explain, including a poleward expansion of

the subtropics, whereE2 P. 0, and a poleward shift of

the E 2 P minimum associated with the midlatitude

storm tracks. Equation (10) also provides limited insight

into the differences in E0 2 P0 across GCMs.

Here we build on HS06’s diffusive interpretation of

hydrologic change in three ways. First, in section 3a, we

use the MEBM to simulate E0 2 P0 in response to a

spatially uniform forcing and feedback. In contrast to

the HS06 approximation, we find that diffusive energy

transport does in fact predict subtropical expansion

and a poleward shift in the storm tracks. These features

of the hydrologic response are found to be directly re-

lated to polar-amplified warming, which the HS06 ap-

proximation does not take into account. In section 3b,

we examine the MEBM response with the full patterns

of forcing, feedbacks, and ocean heat uptake taken from

individual CMIP5 models, and find that the MEBM

captures much of the overall structure of E0 2 P0 in the

ensemble mean, and most of the variability in E0 2 P0

patterns across models. Finally, in section 3c, we use

surface energy constraints to solve for E0 and P0 in-

dependently, and find that these solutions also compare

well with GCMs. Together, these results suggest that dif-

fusive energy transport may provide greater insight into

zonal-meanhydrologic change thanpreviously recognized.

a. Implications of diffusive energy transport forE0 2 P0 in the presence of polar amplification

Let Rf (x) be defined as the TOA radiative forcing

resulting from an abrupt quadrupling of atmospheric

CO2, and G0(x) be defined as the change in ocean heat

uptake. Following the linear feedback framework of

Roe et al. (2015), the perturbation form of the MEBM

can then be expressed as

Rf(x)2G0(x)1 l(x)T 0(x)5

1

2pa2dF 0

dx, (12)

where l(x) is the strength of the local climate feedback,

and F0(x) is the change in atmospheric heat transport,

which in the MEBM is given by

F 0(x)522pp

s

gD(12 x2)

dh0

dx, (13)

where h0(x) 5 cpT0(x) 1 Lq0(x) represents the pertur-

bation near-surface moist static energy.

Because q0 ’ aT0q under constant relative humidity

(according to a linearization of the Clausius–Clapeyron

equation), the solution to Eqs. (12) and (13) will depend

to some extent on the reference climate, which we define

here to be the MEBM solution from reanalysis (Fig. 2,

red lines). As in the mean-state MEBM, we assume that

h0T 5 1:063 h0(0), implying a modest increase in gross

moist stability at the equator. We also assume that D in

Eq. (13) is unchanged from the mean-state diffusion

coefficient. Combining Eqs. (12) and (13) yields a single

differential equation that we solve numerically for T0(x)given Rf (x), G

0(x), and l(x). The solution for E 0 2 P0

then follows fromT0, just as described for themean-state

climate in section 2.

For the first part of this analysis, we assume that G0 50 and that Rf and l are spatially uniform, with values of

8Wm22 and21.5Wm22K21, respectively. These values

are typical of the global means found in simulations of

abrupt CO2 quadrupling, which we discuss in section 3b.

Figure 4a shows the pattern of warming given by the

MEBM in response to this uniform forcing and feed-

back. In agreement with previous studies of the MEBM

response to radiative forcing (Roe et al. 2015), Fig. 4a

shows significantly more warming at high latitudes than

in the tropics. Since Rf, l, and G0 are all spatially uni-

form, such polar amplification of surface warming in the

MEBM can only result from the nonlinearity of the

Clausius–Clapeyron equation, which dictates that water

vapor will increase most in warm regions, causing an

increase in the meridional gradient of latent heat, and

thus an increase in the rate of latent heat transport from

the tropics to the poles.

TheE0 2 P0 profile implied by this pattern of warming

is given in Fig. 4b (solid red line), along with the HS06

approximation from Eq. (10) (blue line), which repre-

sents an amplification of the mean-state E 2 P profile.

Comparing the two curves, three differences stand out:

1) The magnitude of E0 2 P0 is generally weaker in the

MEBM than in theHS06 approximation, particularly

outside the tropics.

2) The MEBM shows an expansion of the subtropical

region where E2 P. 0 (Fig. 4, vertical green lines),

whereas HS06 simply amplifies the existing spatial

pattern of E 2 P.

3) The MEBM places the extratropical minimum of

E0 2 P0 poleward of its climatological position (Fig. 4,

vertical purple lines). This minimum corresponds to

the latitude of maximum moisture convergence, the

poleward shift of which is commonly attributed to a

shift in the latitude of the midlatitude storm tracks.

As we will show, each of these differences stems from

the modification of poleward heat transport by polar

amplification in the MEBM, which HS06 did not take

into account. To explain why, it is convenient to revert to

7486 JOURNAL OF CL IMATE VOLUME 31

the more primitive version of theMEBM in which latent

heat is fluxed downgradient everywhere (Fig. 4b, dashed

red line). While this version gives the wrong hydrologic

cycle in the tropics, its profile of E0 2 P0 is nearly

equivalent to the Hadley cell version of the MEBM

poleward of about 408 latitude, where the HS06 ap-

proximation is most different from the MEBM solution.

In the primitive MEBM, the perturbation northward

latent heat transport is approximately given by

F 0q(x)’bF

q(x) , (14)

where

b5dT 0/dxdT/dx

1

�a2

2

T

�T 0 (15)

represents the fractional change in Fq(x) (see the ap-

pendix for derivation). Then E0 2 P0 is equal to the di-

vergence of F 0q(x), and comprises two distinct terms:

E0 2P0 5b(E2P)

zfflfflfflfflfflffl}|fflfflfflfflfflffl{Term1

11

2pa2Fq

db

dx

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{Term2

. (16)

Equation (14) implies that the profile of latent heat

transport is amplified under global warming by a factor

of b(x). If warming were spatially uniform, Fq(x) would

scale at close to the Clausius–Clapeyron rate of aT0, andE2 Pwould scale at a similar rate because term 2 in Eq.

(16) would be small, owing to weak meridional varia-

tions in a 2 2/T. Thus, if T0 were spatially uniform, the

MEBMsolution forE0 2P0 would be consistent with theapproximation of HS06 [Eq. (10)].

In reality, however, T0 increases with latitude (Fig. 4a),

which impacts the hydrologic cycle in two ways. First, by

weakening the meridional temperature gradient, polar

amplification decreases b everywhere [first term in Eq.

(15)], and thus partially offsets the increase in Fq from the

Clausius–Clapeyron effect. This partly explains the lower

magnitude of E 2 P amplification in the MEBM com-

pared to the HS06 approximation, but does not explain

the poleward shift of E 2 P in the MEBM.

Second, because of theweak temperature dependence

of a, the second term in Eq. (15) increases almost line-

arly with T0, implying a larger value of b (and thus a

larger fractional increase in Fq) at high latitudes. In

terms of the hydrologic cycle, the increase in b with

latitude contributes to an increase in E0 2 P0 every-where, but especially in the midlatitudes where Fq is

greatest [Eq. (16), term 2]. In the subtropics, this positive

contribution to E0 2 P0 results in an expansion of the

region whereE2 P. 0, and thus a poleward shift in the

latitude where E 2 P 5 0. Farther poleward, the mag-

nitude of this positive E0 2 P0 contribution decreases

with latitude in rough proportion to Fq. This implies a

larger (smaller) contribution equatorward (poleward) of

the extratropical E 2 P minimum, and thus a poleward

shift in the latitude where this minimum occurs.

A second perspective on how polar amplification al-

ters the spatial pattern of E 2 P comes from the global

water budget. Like Fq (Table 1), F 0q can be separated

into eddy and Hadley cell contributions, both of which

vanish at the poles. Since E0 2P0 } dF 0q/dx, these

boundary conditions imply that the eddies and Hadley

cell independently satisfy the conservation law thatE 0 5P0 in the global mean.

Now suppose that the HS06 approximation was cor-

rect, such that the fractional change in E 2 P was equal

to aT0 everywhere. Because of polar amplification, the

eddy component ofE0 2P0 in this scenario would have a

FIG. 4. (a) The change in zonal-mean surface temperature in the MEBM in response to a spatially uniform

radiative forcing and feedback. (b) The change inE2 P from theMEBM (red) and the HS06 approximation [blue;

Eq. (10)]. The dashed red line represents theMEBM solution with no HC [i.e.,w(x)5 0]. TheHS06 approximation

is calculated using the MEBMwarming profile from (a), but does not account for the impact of polar amplification

on poleward heat transport. The vertical green lines indicate the highest latitude whereE2P5 0 in themean-state

MEBM solution (Fig. 2f), which is a rough measure of the subtropical boundary. The vertical purple lines indicate

the extratropical minimum in mean-state E 2 P, which is related to the mean-state position of the storm tracks.

15 SEPTEMBER 2018 S I L ER ET AL . 7487

larger magnitude where E0 , P0 (at high latitudes) than

where E0 . P0 (in the subtropics), implying (impossibly)

that E0 6¼ P0 in the global mean. The second term in Eq.

(16) therefore represents a necessary correction to the

HS06 approximation in the presence of polar amplifi-

cation, without which the global water budget would not

close. Because of downgradient energy transport, this

correction will tend to be concentrated in midlatitudes

where Fq is greatest, thereby causing the region over

which E 2 P . 0 to expand, and shifting the latitude of

minimum E 2 P poleward. While these changes in the

hydrologic cycle are often attributed to dynamical

changes within the Hadley cell (Lu et al. 2007) and

midlatitude storm tracks (HS06; Lu et al. 2010), this result

suggests that downgradient energy transport may provide

a complementary—and perhaps more fundamental—

explanation.

b. Connecting intermodel variability in E0 2 P0 to thespatial pattern of climate feedbacks

In GCMs, Rf, G0, and l are not spatially uniform, as

we assumed in section 3a. Roe et al. (2015) showed that

spatial structure in these variables can have a significant

impact on the pattern of surface warming, including the

magnitude of polar amplification. Here we investigate

the effect of such spatial structure on the profile of E0 2P0 predicted by CMIP5 GCMs in response to abrupt

CO2 forcing. To minimize the influence of natural vari-

ability, we focus on years 126–150 after CO2 is quadru-

pled, which represent the last 25 years of most GCM

simulations. However, we have repeated our analysis

using other time periods, with similar results in each case.

For each model, we estimate G0(x) from the net

change in surface energy fluxes, and Rf (x) and l(x) as

the y intercept and slope, respectively, of the regression

line relating the time series of annual-mean surface

temperature to the net radiation at the top of the at-

mosphere (Gregory et al. 2004). While this method may

be less accurate than estimating Rf (x) and l(x) from

fixed SST simulations with andwithout increasedCO2, we

find that the two methods produce very similar results in

all models for which fixed SST simulations are available.

We therefore choose the regression method because it

allows us to include more models in our analysis.2

Figure 5a shows the average profile of Rf (x), G0(x),

and l(x) in the ensemble mean of the CMIP5 simula-

tions, while Figs. 5b and 5c show the resulting patterns of

T0 and E0 2 P0 from both the MEBM (red) and the

GCMs (blue). In Fig. 5c, the vertical green lines indicate

the highest latitudes where E 2 P 5 0 in the pre-

industrial GCM simulations, marking the poleward edge

of the subtropics (Fig. 1). The extratropical minimum of

E2 P is represented by a vertical purple line, which we

interpret as a rough measure of the climatological

storm-track position.

With respect to surface warming (Fig. 5b), theMEBM

solution capturesmuch of the large-scale structure of the

GCM solution, including its hemispheric asymmetry,

though it underpredicts the magnitude of polar ampli-

fication in the Northern Hemisphere by almost 4K at

the pole (14.7 vs 11.0K). Similarly, for E0 2 P0 (Fig. 5c),

FIG. 5. (a) Rf (orange), change in G0 (green; opposite sign), and

l (purple; Wm22K21; as indicated by the right y axis) in the en-

semble mean of 20 CMIP5 simulations of abrupt CO2 quadrupling.

(b) The zonal-mean profile of surface warming averaged over years

126–150 after CO2 quadrupling in the CMIP5 ensemble mean (blue)

and theMEBM(red). (c)The change inE2P in theCMIP5ensemble

mean (blue) and the MEBM (red). The dashed red line represents

the HS06 approximation. The vertical green (purple) lines indicate

the highest latitude where E2 P5 0 (E2 P has a local minimum)

in the preindustrial GCM simulations.

2 The names of the models included in our analysis are listed

above the panels in Fig. 6.

7488 JOURNAL OF CL IMATE VOLUME 31

the MEBM captures much of the overall response in the

GCMs, including a poleward shift in both the sub-

tropical boundary (Fig. 5c, green lines) and the extra-

tropicalE2Pminimum (Fig. 5c, purple lines). But here,

too, the MEBM is imperfect: in the Southern Hemi-

sphere, in particular, the magnitude of the poleward

shift is too weak, suggesting that diffusion of near-

surface moist static energy does not by itself explain

the total change in zonal-mean E2 P in this region. Yet

compared with theHS06 approximation (Fig. 5c, dashed

line), theMEBMsolution represents a clear improvement,

highlighting the importance of spatial variations in Rf, l,

G0, and T0 in determining the response of the zonal-mean

hydrologic cycle to global warming.

When the full ensemble of CMIP5 simulations is

considered (Fig. 6), it is clear that the spatial patterns of

Rf,G0, and l affect the detail of the climate response, but

not the main features (e.g., polar amplification, wetter

deep tropics, drier subtropics, and wetter high latitudes),

indicating a high degree of agreement among models in

these basic responses. Even so, the MEBM captures

much of the intermodel variability in both T0 (Fig. 6,top) and E0 2 P0 (Fig. 6, bottom) in response to differ-

ences in Rf, G0, and l. A quantitative assessment of the

FIG. 6. As in Figs. 5b,c, but for each of the 20 CMIP5 ensemblemembers included in our analysis. (top) Surface warming and (bottom) the

change in E 2 P.

15 SEPTEMBER 2018 S I L ER ET AL . 7489

MEBM’s skill is presented in Fig. 7, which shows the

correlation coefficient between the GCM and MEBM

values of T0 (red) and E0 2 P0 (blue) as a function of

latitude. The dashed black line represents the threshold

for statistical significance at the 95% confidence level

(r5 0.44).WithT0 (Fig. 7, red), we find high correlationsat all latitudes, with a global-mean value of r 5 0.90.

Although the correlations are somewhat weaker in polar

regions of the Northern Hemisphere (r ’ 0.6–0.8), they

remain well above the threshold of statistical signifi-

cance everywhere. In the case of E0 2 P0 (Fig. 7, blue),the correlations are also strong in most regions (r5 0.61

in the global mean), and are statistically significant over

87% of the globe. These results imply that much of the

intermodel differences in the zonal-mean response of

the hydrologic cycle to global warming can be explained

by the principle of downgradient energy transport ap-

plied to different patterns of forcing, feedbacks, and

ocean heat uptake.

c. Partitioning E0 2 P0 into E0 and P0

Because the MEBM only simulates energy transport

and convergence, its solution forE0 2P0 implies nothing

about P0. However, if E0 can be determined in-

dependently, the solution forP0 can then be found as thedifference between E0 and E0 2 P0.In their paper, HS06 assumed that E0 was simply

proportional toE, with a spatially uniform scaling factor

equal to the fractional change in global-mean evapora-

tion with global warming (;2%K21). This approxima-

tion is shown in Fig. 8 (dashed line), along with the

actual profile ofE0 taken from the ensemble mean of the

CMIP5 simulations (blue line). Comparing the two

curves, we find that the HS06 approximation captures

the overall structure and magnitude of E0 reasonably

well, but is too large in the tropics, and misses almost all

of the regional-scale variations found in the GCM

simulations.

A more accurate estimate of E0 can be obtained us-

ing the surface energy approach of Siler et al. (2018).

Building on earlier work by Penman (1948) andMonteith

(1981), Siler et al. (2018) combine the surface energy

budget with the bulk transfer equations for latent and

sensible heat flux to arrive at a single expression for the

change in surface evaporation with warming. Over the

FIG. 7. The Pearson correlation coefficient at each latitude be-

tween GCM and MEBM solutions across the 20 CMIP5 ensemble

members: T0 (red line), E0 2 P0 (blue line), and the threshold for

statistical significance at the 95% confidence level (dashed line; r50.44, 18 degrees of freedom).

FIG. 8. (a) The change in zonal-mean evaporation in the CMIP5

ensemble mean (blue) and the MEBM (red), given by Eq. (17)

assuming R0s 5 0. The dashed red line represents the HS06 ap-

proximation. (b) As in (a), but for zonal-mean precipitation,

which was computed by subtracting the profiles ofE0 2P0 in Fig. 5c

from the profiles of E0 in (a). (c) The fractional change in pre-

cipitation per degree of local surface warming. The solid gray line

represents Clausius–Clapeyron scaling [Eq. (11)], and the dashed

gray line at 2%K21 approximates the global-mean change pre-

dicted by GCMs.

7490 JOURNAL OF CL IMATE VOLUME 31

oceans, wheremost global evaporation occurs, the change

in evaporation can be approximated as

E0 ’ET 0b

0(a2 2/T)1R0

s 2G0

11b0

, (17)

where R0s is the change in net radiation at the surface,

b0(T) is the Bowen ratio in the limit of a saturated near-

surface atmosphere, and all other variables are as de-

fined previously. In the same CMIP5 simulations we

analyze here, Siler et al. (2018) found that the first term

on the rhs of Eq. (17) was the most important globally,

whileG0 was identified as a leading control on the spatialpattern of E0. Meanwhile, R0

s was found to play a sec-

ondary role on both regional and global scales, and thus

we choose to neglect it here. For E, we use the CMIP5

ensemble mean. All other variables in Eq. (17) can be

determined from existing input (G0) or output (T0 andT),allowing us to solve for E0 (and thus P0) without addingany variables or complexity to the MEBM.

The red line in Fig. 8a shows the profile of E0 given by

Eq. (17), with R0s 5 0. Compared with the HS06 ap-

proximation (dashed line), Eq. (17) is more accurate at

most latitudes, and also captures many of the regional

variations in E0 that are absent from the HS06 approx-

imation. Taking the difference between E 0 and E 0 2 P0

(Fig. 5c) yields P0, which is shown in Fig. 8b for the

CMIP5 ensemble (blue), the HS06 approximation

(dashed line), and our MEBM solution (red line). As

with E 0 and E 0 2 P0, the MEBM solution for P0 rep-resents a substantial improvement over the HS06 ap-

proximation at most latitudes. This is particularly true in

the tropics, where the MEBM successfully captures the

large increase in precipitation near the equator, consis-

tent with a narrowing of the ITCZ that has been at-

tributed in part to the enhanced pole–equator gradient

in moist static energy (Byrne and Schneider 2016).

Elsewhere, the MEBM shows a decrease in subtropical

precipitation and a large increase at high latitudes,

which is also broadly consistent with GCM simulations.

This agreement suggests that projected changes in

zonal-mean precipitation are largely a consequence of

constraints on atmospheric energy transport and surface

fluxes, modulated by spatial variations in radiative

forcing, feedbacks, and ocean heat uptake.

Figure 8c shows the same precipitation changes as

Fig. 8b, but as a percentage change per degree of local

warming. The gray dashed line in Fig. 8c represents the

global-mean increase of 2%K21, while the solid gray

line represents the larger Clausius–Clapeyron rate of

increase in atmospheric water vapor under constant

relative humidity [a; Eq. (11)]. The MEBM emulates

well the overall magnitude and pattern of the sensitivity

of the GCM ensemble mean, albeit with notable mis-

matches in the subtropics and Southern Hemisphere

midlatitudes. In the deep tropics and high latitudes, in-

creases in latent heat flux convergence drive increases in

precipitation that significantly exceed the global-mean

sensitivity of 2%K21, approaching or even exceeding

the Clausius–Clapeyron rate at some latitudes. In the

subtropics, meanwhile, the change in precipitation is

small and/or negative, allowing the global-mean value

of 2%K21 to be maintained. Significantly, this spa-

tial pattern of precipitation sensitivity in the MEBM

does not require knowledge of the specific mechanisms

(e.g., stratiform or convective processes) generating

the precipitation; everywhere, the precipitation change

is simply the result of global energetic and transport

constraints.

4. Discussion

In this paper, we have incorporated a new Hadley cell

parameterization into an existing moist static energy

balance model (MEBM), and used it to investigate the

implications of downgradient energy transport for the

response of the zonal-mean hydrologic cycle to global

warming. In the idealized case of spatially uniform ra-

diative forcing and feedbacks, we found that down-

gradient energy transport implies a poleward expansion

of the subtropics, whereE2 P. 0, and a poleward shift

in the extratropical minimum of E 2 P, which is con-

sistent with a change in storm-track latitude.We showed

that both of these phenomena are tied to polar-amplified

warming in the MEBM, which is itself closely tied to

downgradient energy transport (Roe et al. 2015). When

forced with realistic patterns of forcing, feedbacks, and

ocean heat uptake from comprehensive GCMs, the

MEBM was found to broadly replicate the simulated

changes in E 2 P, while also capturing much of the in-

termodel variability across the CMIP5 ensemble. Given

the surface energy constraints on evaporation change,

the MEBM predicts changes in zonal-mean pre-

cipitation that are also consistent withGCMprojections,

including a decrease in subtropical precipitation and a

contraction and intensification of the ITCZ.

There are surely ways to make the MEBM more

realistic—for example, by allowing D to vary with lati-

tude or climate state, or adjusting w(x) to include Ferrel

and polar cells. However, in experimenting with such

modifications, we have found that their impact onE0 2 P0

is generally small. For example, when we increase D by

20% in the global warming simulation,E0 2P0 changes byless than 2.5Wm22 at all latitudes. Similarly, when we

add a Ferrel cell to theMEBM [w(x)5 cos(px)(12 jxj)],the change inE0 2 P0 is less than 3Wm22 everywhere. In

15 SEPTEMBER 2018 S I L ER ET AL . 7491

both cases, the changes result in better agreement with

GCMs in the Southern Hemisphere midlatitudes, but the

improvement is modest relative to existing differences of

about 10Wm22 between theMEBMandGCM solutions

in this region (Fig. 5c).

Another question to consider is the extent to which

the spatial patterns of local feedbacks l and the change

in ocean heat uptake G0 are truly independent of at-

mospheric heat and moisture transport, as we have as-

sumed here. While the spatial pattern of G0 is set by

regional ocean circulations (Marshall et al. 2015;

Armour et al. 2016), it seems plausible that its magni-

tude may be, in part, set by the magnitude of local at-

mospheric heat flux convergence. Likewise, there is

evidence that l depends on the spatial pattern of surface

warming within GCMs (Rose et al. 2014; Andrews and

Webb 2018; Po-Chedley et al. 2018). An interesting

avenue of future research would be to evaluate these

interactions with the MEBM coupled to a dynamic

ocean while allowing for local feedbacks to vary with the

pattern of warming.

Finally, our results raise interesting questions about

the influence of atmospheric dynamical changes on the

zonal-mean hydrologic cycle. From mixing length

theory, our assumption of fixed diffusivity D is equiv-

alent to assuming that changes in eddy dynamics are

negligible (e.g., Frierson et al. 2007). Moreover, as

implemented here [Eq. (6)], the MEBM does not ac-

count for possible changes in the partitioning between

eddy and Hadley cell energy transport that might re-

sult from dynamical changes. Yet despite these sim-

plifications, the MEBM gives changes in E 2 P that

are consistent with poleward shifts in the storm tracks

and subtropics—responses that have previously been

attributed to dynamical changes in the eddies and

Hadley cell (e.g., HS06; Lu et al. 2007; Scheff and

Frierson 2012). Further study is needed to better under-

stand the relationship between such dynamical changes

and downgradient energy transport (e.g., Tandon et al.

2013; Mbengue and Schneider 2017, 2018). Even with-

out invoking dynamical changes, however, our study

shows that the principle of downgradient energy trans-

port provides a simple explanation for many aspects of

the zonal-mean hydrologic cycle and its simulated re-

sponse to global warming.

Acknowledgments. We are grateful to Spencer Hill

and two anonymous reviewers for their excellent com-

ments. We also acknowledge the important contribution

of Marcia Baker, who first proposed the idea of modi-

fying the MEBM to include a realistic hydrologic cycle.

KCA’s contribution was funded by the National Science

Foundation (AGS-1752796).

APPENDIX

Estimating the Change in Diffusive Latent HeatTransport with Warming

In the original MEBM, latent heat is transported

downgradient with the same diffusivity as total moist

static energy [Eq. (2)]:

Fq(x)52

2pps

gD(12 x2)L

dq

dx. (A1)

According to the Clausius–Clapeyron equation

dq

dT5aq , (A2)

the meridional moisture gradient in Eq. (A1) is strongly

dependent on the meridional temperature gradient

dq

dx5aq

dT

dx. (A3)

Substituting Eq. (A3) into Eq. (A1) and differentiating

yields the change in northward latent heat flux

F 0q(x)’2

2pps

gD(12 x2)L

�T 0d(aq)

dT

dT

dx1aq

dT 0

dx

�,

(A4)

which simplifies to Eq. (14).

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